1. 07 - constitutive equations
density growth
07 - constitutive equations

2. constitutive equations
constitutive equations in structural analysis, constitutive rela-tions connect applied stresses or forces to strains or deformations. the constitutive relations for linear materials are linear. more generally, in physics, a constitutive equation is a relation between two physical quantities (often tensors) that is specific to a material, and does not follow directly from physical law. some constitutive equations are simply phenomenological; others are derived from first principles.
constitutive equations

3. constitutive equations
or equations of state bring in the charac-terization of particular materials within continuum mechanics. mathematically, the purpose of these relations is to supply connections between kinematic, mechanical and thermal fields. physically, constitu-tive equations represent the various forms of idealized material response which serve as models of the behavior of actual substances.
Chadwick „Continuum mechanics“ [1976]
constitutive equations

4. constitutive equations
tensor analysis - basic derivatives
constitutive equations

5. constitutive equations
tensor analysis - basic derivatives
constitutive equations

6. constitutive equations
neo hooke‘ian elasticity
• free energy
• definition of stress
• definition of tangent operator
constitutive equations

7. constitutive equations
neo hooke‘ian elasticity
• free energy
• definition of stress
• definition of tangent operator
constitutive equations

8. constitutive equations
neo hooke‘ian elasticity
undeformed
potato
deformed
potato
• free energy
• definition of stress
constitutive equations

9. constitutive equations
neo hooke‘ian elasticity
undeformed
potato
mashed
potatoes
• free energy
• definition of stress
• remember! mashing potatoes is not an elastic process!
constitutive equations

10. constitutive equations
neo hooke‘ian elasticity
undeformed
potato
deformed
potato
• free energy
• large strain - lamé parameters and bulk modulus
• small strain - young‘s modulus and poisson‘s ratio
constitutive equations

11. constitutive equations
neo hooke‘ian elasticity in cellular materials
free energy
Carter & Hayes [1977]
constitutive equations

12. constitutive equations
density growth at constant volume
• free energy
• stress
• mass flux
• mass source
constitutive coupling of growth and deformation
Gibson & Ashby [1999]
constitutive equations

13. constitutive equations
density growth - mass source
1d model problem
gradually increased workout
how does the
mass source control growth?
constitutive equations

14. constitutive equations
density growth - mass source
resorption
growth
increasing forces causes density increase
constitutive equations

15. constitutive equations
density growth - mass source
first time interval
homog.
increasing force causes energy increase
constitutive equations

16. constitutive equations
density growth - mass source
convergence towards biological equilibrium
constitutive equations

17. constitutive equations
density growth - mass source
parameter sensitivity wrt
constitutive equations

18. constitutive equations
density growth - mass source
parameter insensitivity wrt
constitutive equations

19. constitutive equations
density growth - mass source
the density develops such that the tissue can
just support the given mechanical load
constitutive equations

20. constitutive equations
density growth - mass flux
initial hat type density distriubtion
whatz this
mass flux good for
in the end?
constitutive equations

21. constitutive equations
density growth - mass flux
initial hat type density distriubtion
„…mon dissin ne représentait pas un chapeau. il représen-tait un serpent boa qui digérait un éléphant. j‘ai alors dessiné l‘intérieur du serpent boa, afin que les grandes personnes puissent comprendre. elles ont toujours
besoin d‘explications…“
Antoine de Saint-Exupéry, Le Petit Prince [1943]
constitutive equations

22. constitutive equations
density growth - mass flux
equilibration of concentrations
constitutive equations

23. constitutive equations
density growth - mass flux & source
smoothing influence of mass flux
constitutive equations

24. example - bone loss in space
density growth - bone loss in space
human spaceflight to mars could become a reality within the next 25 years, but not until some physiological problems are resolved, including an alarming loss of bone mass, fitness and muscle strength. gravity at mars' surface is about 38 percent of that on earth. with lower gravi-tational forces, bones decrease in mass and density. the rate at which we lose bone in space is times greater than that of a post-menopausal woman and there is no evidence that bone loss ever slows in space. further, it is not clear that space travelers will regain that bone on returning to gravity. druing a trip to mars, lasting between
13 and 30 months, unchecked bone loss could make an
astronaut's skeleton the equivalent of a 100-year-old person.
example - bone loss in space

25. example - bone loss in space
density growth - bone loss in space
nasa has collected data that humans in space lose bone mass at a rate of so far, no astronauts have been in space for more than 14 months but the predicted rate of bone loss seems constant in time. this could be a severe problem if we want to send astronauts on a 3 year trip to mars and back. how long could an astronaut survive in a zero-g environment if we assume the critical bone density to be ? you can assume an
initial density of !
example - bone loss in space

26. example - bone loss in space
density growth - bone loss in space
example - bone loss in space

27. example - bone loss in space
density growth - bone loss in space
Carter & Hayes [1977]
example - bone loss in space

28. example - asymmetric bone mass
density growth - asymmetric bone mass
inter-arm asymmetry
postmenopausal tennis players,who started their participation in tennis after menarche show greater bone size and mass in the loaded arm. the degree of inter-arm asymmetry in bone mass and bone size is proportional to the length of tennis participation.
Sanchis-Moysi et al. [2004]
example - asymmetric bone mass